Get or display normal vector / coordinates of selected vertex?
Unfortunately i dont know a way to display the normal vector in the viewport.
Here are two options to get the normal  vector. I'd say it depends on what you want to do with the normal vector after getting it...

Option 1 Python:

import bpy

mesh = bpy.context.object.data

bpy.ops.object.mode_set(mode='OBJECT')

selected_verts = [v for v in mesh.vertices if v.select]

for v in selected_verts:    
    print(f"Vertex Normal: {v.normal}")


Option 2 Add-On Animation Nodes:


I filtered all the normals with a vertex group I did manually.

additional: here are some answers for modifiing the ui to display normal-information:
Display values of a normal of a selected face